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Prove that the operation is associative
Given:
\( a * b = \frac{ab}{7}, \quad a,b \in \mathbb{Q} \)
Proof:
LHS:
\( (a*b)*c = \left(\frac{ab}{7}\right)*c \)
\( = \frac{\frac{ab}{7} \cdot c}{7} = \frac{abc}{49} \)
RHS:
\( a*(b*c) = a*\left(\frac{bc}{7}\right) \)
\( = \frac{a \cdot \frac{bc}{7}}{7} = \frac{abc}{49} \)
Thus:
\( (a*b)*c = a*(b*c) \)
Conclusion:
✔ Therefore, the operation is associative on \( \mathbb{Q} \).